In this work, we investigated the dip-coating flow generated by a plate lifted at constant speed through a stratified bath made of two immiscible liquids, focusing on the case when both of them are entrained on the plate.
We presented first a general formulation within the framework of the lubrication approximation that includes the full (nonlinear) expression for the curvature and the effect of gravity.
Following a similar scheme as [1]}, we then simplified the problem for small capillary numbers \(Ca^{1/3} \ll 1\) .
Using this asymptotic formulation, we explored the effect of some of the control parameters on the asymptotic thicknesses of the resulting thin liquid films.
We mostly focused our attention on the viscosity ratio \(M\)  and the thickness of the floating layer \(\Delta H\) , as these parameters could be varied more easily in experiments.
For completeness we also showed results for a limited number of values of the surface tension ratio \(\Sigma \) , although this parameter would be harder to vary experimentally using common liquids.
